By Koen Thas

ISBN-10: 3764361581

ISBN-13: 9783764361587

ISBN-10: 3764377976

ISBN-13: 9783764377977

In this monograph finite generalized quadrangles are categorized through symmetry, generalizing the distinguished Lenz-Barlotti class for projective planes. The booklet is self-contained and serves as creation to the combinatorial, geometrical and group-theoretical techniques that come up within the class and within the normal idea of finite generalized quadrangles, together with automorphism teams, elation and translation generalized quadrangles, generalized ovals and generalized ovoids, span-symmetric generalized quadrangles, flock geometry and estate (G), regularity and nets, break up BN-pairs of rank 1, and the Moufang property.

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**Extra info for Symmetry in Finite Generalized Quadrangles**

**Sample text**

In order to study the generalized quadrangles which have some distinct axes of symmetry through some point p, we will introduce “Property (T)”. TGQ’s which satisfy Property (T) for some ordered ﬂag always have order (s, s2 ) for some s, and every TGQ of order (s, t) which has a (proper) subGQ of order s through the translation point satisﬁes Property (T) for some ordered ﬂag(s) containing the translation point. Property (T) is closely related to Property (G), and seems to be more general in the case of translation generalized quadrangles.

1 Chapter 3. The Known Generalized Quadrangles TGQ’s which Arise from Flocks (and their Translation Duals) Kantor semifield flock generalized quadrangles A ﬂock is called linear if all the ﬂock planes contain a common line. In that case, it can be shown that the ﬂock GQ is classical, and conversely. Let K be the quadratic cone with equation X0 X1 = X22 of PG(3, q), q odd. Then the q planes πt with equation tX0 − mtσ X1 + X3 = 0, where t ∈ GF(q), m is a given non-square in GF(q) and σ a given automorphism of GF(q), deﬁne a ﬂock F of K; see [120].

Thas [120]; ﬁrst given in explicit q-clan form in [81]) which is described in detail in [50]. However, it is rather involved and not as elegant as the BLT representation of T. Penttila in [98], which we will give here. Let K = GF(q) ⊆ GF(q 2 ) = K , q odd and q ≥ 5. Let η be a primitive element ¯, of K , and put η = ζ q−1 (so η has multiplicative order q + 1). Put T (x) = x + x where x ¯ = xq , and consider V = {(x, y, a) x, y ∈ K , a ∈ K} as a 5-dimensional vector space over K. Deﬁne a map Q : V → K by Q(x, y, a) = xq+1 + y q+1 − a2 .

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